Semiparametric Estimation in Triangular System Equations with Nonstationarity

A system of multivariate semiparametric nonlinear time series models is studied with possible dependence structures and nonstationarities in the parametric and nonparametric components. The parametric regressors may be endogenous while the nonparametric regressors are assumed to be strictly exogenous. The parametric regressors may be stationary or nonstationary and the nonparametric regressors are nonstationary integrated time series. Semiparametric least squares (SLS) estimation is considered and its asymptotic properties are derived. Due to endogeneity in the parametric regressors, SLS is not consistent for the parametric component and a semiparametric instrumental variable (SIV) method is proposed instead. Under certain regularity conditions, the SIV estimator of the parametric component is shown to have a limiting normal distribution. The rate of convergence in the parametric component depends on the properties of the regressors. The conventional √ n rate may apply even when nonstationarity is involved in both sets of regressors. Existing studies show that both nonstationarity and nonlinear-ity are common features of much economic data. Modeling such data in a way that allows for possible nonstationarity helps to avoid dependence on stationarity assumptions and mixing conditions for all of the variables in the system. At present there is a large literature on parametric linear modeling of nonstationary time series and interest has primarily focused on time series with a unit root or near unit root structure (for an overview, see, for example Phillips and Xiao, 1998, and the references therein). In practical work, much attention is given to multivariate systems and cointegration models. Inferential methods for these linear systems include both parametric and semiparametric (e.g., Phillips, 1995, 1998, forthcoming) approaches. In comparison with work on linear parametric models, there have been only a few studies of parametric nonlinear models with integrated variables. introduced techniques for developing asymptotics for certain classes of nonlinear nonstationary parametric systems and aspects of this work have been extended by Pötscher (2004), Jeganathan (2004, 2008), and Berkes and Horváth (2006). Interest has also developed in nonparametric modeling methods to deal with nonlinearity of unknown form involving nonstationary variables. The last paper examines in a nonparametric setting spurious time series models of the type for which the asymptotic theory was Among nonparametric studies of nonstationarity, two different mathematical approaches have been developed. In one approach, a so-called ''Markov splitting technique'' has been used in Karlsen and Tjøstheim (2001), and Karlsen et al. (2007) to model univariate time series with a null-recurrent structure; …